Contents
1. Notation
| Symbol | Meaning |
|---|---|
| $i$, $d$, $\delta$ | Effective interest rate, effective discount rate and force of interest |
| $v=(1+i)^{-1}$ | Annual discount factor |
| $\ell_x$ | Number of lives at age $x$ in the mortality table |
| $_tp_x$, $_tq_x$ | Probability of surviving / dying within $t$ years from age $x$ |
| $A_x$, $\ddot a_x$ | APV of a whole life insurance and of a whole life annuity-due |
| $P$, $_tV$ | Level net premium and reserve at time $t$ |
2. Theory of interest
$$ (1+i) = (1+\tfrac{i^{(m)}}{m})^{m} = e^{\delta}, \qquad v=\frac{1}{1+i}, \qquad d = 1-v = iv $$
Equivalence between the effective rate, the nominal rate compounded $m$ times a year and the force of interest $\delta$.
$$ a_{\overline{n}|} = \frac{1-v^{n}}{i}, \qquad \ddot a_{\overline{n}|} = \frac{1-v^{n}}{d}, \qquad s_{\overline{n}|} = \frac{(1+i)^{n}-1}{i} $$
Present value of an annuity-immediate and an annuity-due; future (accumulated) value of an annuity.
$$ (Ia)_{\overline{n}|} = \frac{\ddot a_{\overline{n}|} - n v^{n}}{i} $$
Increasing annuity: payments of 1, 2, …, $n$.
3. Mortality and survival
$$ {}_{t}p_x = \frac{\ell_{x+t}}{\ell_x}, \qquad {}_{t}q_x = \frac{\ell_x-\ell_{x+t}}{\ell_x}, \qquad {}_{t|}q_x = {}_{t}p_x\, q_{x+t} $$
Survival, mortality and deferred mortality (death in year $t{+}1$).
$$ \mu_x = -\frac{1}{\ell_x}\frac{d\ell_x}{dx}, \qquad {}_{t}p_x = \exp\!\left(-\int_0^{t}\mu_{x+s}\,ds\right) $$
Force of mortality $\mu_x$ and its relationship with continuous survival.
4. Life contingencies
$$ A_x = \sum_{k=0}^{\infty} v^{k+1}\,{}_{k|}q_x, \qquad \ddot a_x = \sum_{k=0}^{\infty} v^{k}\,{}_{k}p_x $$
Whole life insurance (paid on death) and whole life annuity-due (paid while alive).
$$ A_x = 1 - d\,\ddot a_x, \qquad A^{1}_{x:\overline{n}|} + A_{x:\overline{n}|}{}^{\;1} = A_{x:\overline{n}|} $$
Insurance-annuity relationship and decomposition of the endowment insurance into its death and survival components.
5. Net premiums
$$ P(A_x) = \frac{A_x}{\ddot a_x} = \frac{1}{\ddot a_x} - d $$
Level net premium for whole life insurance, under the equivalence principle.
$$ P_{x:\overline{n}|} = \frac{A_{x:\overline{n}|}}{\ddot a_{x:\overline{n}|}} $$
Net premium for an $n$-year endowment insurance.
6. Policy reserves
$$ {}_{t}V = A_{x+t} - P(A_x)\,\ddot a_{x+t} $$
Prospective reserve: present value of future benefits minus present value of future premiums.
$$ {}_{t}V = \frac{A_{x+t}-A_x}{1-A_x} = 1 - \frac{\ddot a_{x+t}}{\ddot a_x} $$
Equivalent forms of the whole life insurance reserve.
7. Risk theory
$$ S = \sum_{i=1}^{N} X_i, \qquad E[S]=E[N]\,E[X], \qquad \operatorname{Var}(S)=E[N]\operatorname{Var}(X)+\operatorname{Var}(N)\,E[X]^2 $$
Collective risk model: aggregate claims and their variance (compound distribution formula).
$$ \Pi = E[S] + \theta\,\sigma_S \quad\text{(standard deviation loading)}, \qquad \psi(u)\le e^{-Ru} \;\text{(Lundberg inequality)} $$
Premium with a safety loading and the bound on the probability of ruin $\psi(u)$ with adjustment coefficient $R$.
Want the conceptual background? Read Actuarial Mathematics. Where are these taught? In the courses of the degree program.